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A096309 a(1)=1; for n > 1, a(n) is the number of levels in the "stacked" prime number factorization of n (prime number factorization of the exponents if necessary and so on ...).

%I A096309 #14 Nov 18 2017 20:59:36
%S A096309 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,2,1,1,
%T A096309 1,2,1,1,1,2,1,1,1,2,2,1,1,3,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,2,1,1,1,2,
%U A096309 1,1,1,2,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,2,2,2
%N A096309 a(1)=1; for n > 1, a(n) is the number of levels in the "stacked" prime number factorization of n (prime number factorization of the exponents if necessary and so on ...).
%C A096309 For n > 1: a(n)=1 iff n squarefree.
%C A096309 Sequence A185102 is a (better?) variant, identical except for A185102(1)=0. - _M. F. Hasler_, Nov 21 2013
%H A096309 Antti Karttunen, <a href="/A096309/b096309.txt">Table of n, a(n) for n = 1..65537</a>
%e A096309 a(4)=2 because 4=2^2; a(8)=2 because 8=2^3; a(16)=3 because 16=2^(2^2).
%e A096309 a(65536) = a(2^2^2^2) = a(2^^4) = 4 is the first term larger than 3; the index of the first a(n) > 4, n = 2^^5, has 19729 digits. - _M. F. Hasler_, Nov 21 2013
%t A096309 f[n_Integer] := FactorInteger[n][[All, 2]]; a[n_] := Depth[f[n] //. k_Integer /; k > 1 :> f[k]] - 1; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Nov 20 2013 *)
%o A096309 (PARI) A096309=n->if(n>1,vecmax(apply(a,factor(n)[,2])))+1 \\ _M. F. Hasler_, Nov 21 2013
%Y A096309 Cf. A087049, A185102.
%K A096309 easy,nonn
%O A096309 1,4
%A A096309 _Franz Vrabec_, Jun 27 2004
%E A096309 More terms from _Jean-François Alcover_, Nov 20 2013